**Part 3**

**The Solar Wind Dynamics – From Large to Small Scales** 

140 Exploring the Solar Wind

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**0**

**7**

*Poland*

Wiesław M. Macek

**Multifractal Turbulence in the Heliosphere**

*and Space Research Centre, Polish Academy of Sciences*

*Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszy ´nski University*

The aim of the chapter is to give an introduction to the new developments in turbulence using nonlinear dynamics and multifractals. To quantify scaling of turbulence we use a generalized two-scale weighted Cantor set (Macek & Szczepaniak, 2008). We apply this model to intermittent turbulence in the solar wind plasma in the inner and the outer heliosphere at the ecliptic and at high heliospheric latitudes, and even in the heliosheath, beyond the termination shock. We hope that the generalized multifractal model will be a useful tool for analysis of intermittent turbulence in the heliospheric plasma. We thus believe that multifractal analysis of various complex environments can shed light on the nature of

Nonlinear dynamical systems are often highly sensitive to initial conditions resulting in chaotic motion. In practice, therefore, the behavior of such systems cannot be predicted in the long term, even though the laws of dynamics unambiguously determine its evolution. Chaos is thus a non-periodic long-term behavior in a deterministic system that exhibits sensitivity to initial conditions. Yet we are not entirely without hope here in terms of predictability, because in a dissipative system (with friction) the trajectories describing its evolution in the space of system states asymptotically converge towards a certain invariant set, which is called an attractor; strange attractors are fractal sets (generally with a fractal dimension) which exhibit

We remind that a fractal is a rough or fragmented geometrical object that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Strange attractors are often fractal sets, which exhibits a hidden order within chaos. Fractals are generally *self-similar* and independent of scale (generally with a particular fractal dimension). A multifractal is an object that demonstrate various self-similarities, described by a multifractal spectrum of dimensions and a singularity spectrum. One can say that self-similarity of multifractals is point dependent resulting in the singularity spectrum. A multifractal is therefore in a certain sens like a set of intertwined fractals

Starting from seminal works of Kolmogorov (1941) and Kraichnan (1965) many authors have attempted to recover the observed scaling laws, by using multifractal phenomenological

**1. Introduction**

turbulence.

**1.1 Chaos and fractals basics**

(Macek & Wawrzaszek, 2009).

**1.2 Importance of multifractality**

a hidden order within the chaos (Macek, 2006b).
